290 Dr. J. W. Nicholson on the Bending of 



Using these values, after some reduction, we find 



where 



^ _ ^ V 4? 12?' ' ' ( ° 

 and the sum of the series becomes 



*=ffi{> + *}***> ■ ■ ■ < 138) 



all values being taken at the zero point. 



The analysis of this section and that immediately pre- 

 ceding does not apply to the present problem alone. It is 

 generally applicable to all similar problems in which a 

 second approximation is desired in regions of space for 

 which incident and reflected waves are concerned, provided 

 that an harmonic series may be obtained. 



Application to the special case. 



We proceed to apply these results to the special problem 

 in hand. In this problem, in the region of brightness, as in 

 an earlier section, 



zv = (f> n — (j) nr —m0, (139) 



where 6 is replaced by if the Mehler Dirichlet integral 

 must be used. But when points not close to the axis are 

 concerned, the important harmonics are those of order z, for 

 the effect depends upon the values of the various functions, 



by the formulae just proved, at the zero point, which is given 

 by # = rsin#/R in the notation of the figure, and this corre- 

 sponds to a value of m of order z. The zonal harmonic may 

 therefore be expanded asymptotically, if sufficient terms are 

 retained. The usual formula of course fails, and we proceed 

 to obtain the necessary modification. 



